A Mathematical Framework of Semantic Communication based on Category Theory

Abstract: While semantic communication (SemCom) has recently demonstrated great potential to enhance transmission efficiency and reliability by leveraging ML and knowledge base (KB), there is a lack of mathematical modeling to rigorously characterize SemCom system and quantify the performance gain obtained from ML and KB. In this paper, we develop a mathematical framework for SemCom based on category theory, rigorously modeling the concepts of semantic entities and semantic probability space. Within this framework, we introduce the semantic entropy to quantify the uncertainty of semantic entities. We theoretically prove that semantic entropy can be effectively reduced by exploiting KBs, which capture semantic dependencies. Within the formulated semantic space, semantic entities can be combined according to the required semantic ambiguity, and the combined entities can be encoded based on semantic dependencies obtained from KB. Then, we derive semantic channel capacity modeling, which incorporates the mutual information obtained in KB to accurately measure the transmission efficiency of SemCom. Numerical simulations validate the effectiveness of the proposed framework, showing that SemCom with KB integration outperforms traditional communication in both entropy reduction and coding efficiency.